Answer:
12 days.
Step-by-step explanation:
[tex]\because Efficiency = \frac{Work}{Time}[/tex]
Let x be the time taken by machine Y to produce w widget,
So, the efficiency of machine Y = [tex]\frac{w}{x}[/tex],
Since, X takes 2 days longer to produce w widgets than Machine Y,
So, the efficiency of machine X = [tex]\frac{w}{x+2}[/tex],
If total work = [tex]\frac{5}{4}w[/tex],
Then the time taken by machines X and Y when they work together
[tex]=\frac{5/4w}{\text{efficiency of machine A}+\text{efficiency of machine B}}[/tex]
[tex]=\frac{5w}{4(\frac{w}{x+2}+\frac{w}{x})}[/tex]
[tex]=\frac{5}{4(\frac{x+x+2}{x(x+2)})}[/tex]
[tex]=\frac{5x(x+2)}{8x+8}[/tex]
According to the question,
[tex]\frac{5x(x+2)}{8x+8}=3[/tex]
[tex]5x^2+10x = 24x+24[/tex]
[tex]5x^2-14x-24=0[/tex]
[tex]5x^2-20x+6x-24=0[/tex]
[tex]5x(x-4)+6(x-4)=0[/tex]
[tex](5x+6)(x-4)=0[/tex]
[tex]\implies 5x+6=0\text{ or }x-4=0[/tex]
[tex]\implies x=-\frac{6}{5}\text{ or }x=4[/tex]
Number of days can not be negative,
Hence, the time taken by machine X to produce w widgets = x + 2 = 4 + 2 = 6 days,
Therefore the time taken by machine X to produced 2w widgets would be 12 days.
